While constructing anomaly-free string theories with $\mathcal N=1$ supersymmetry (16 supercharges constituting a Majorana-Weyl spinor), we learn that the gauge group must be 496-dimensional in order to cancel some terms (like $\mathrm{tr} R^6$) in the anomaly polynomial, which is a 12-form characteristic class. There are also some other more involved conditions that I will not reproduce here.
This is satisfied by the groups $E_8\times E_8$ and $\mathrm{SO}(32)$, for which the corresponding heterotic string theories are very well-known. However, the groups $E_8\times U(1)^{248}$ and $U(1)^{496}$ are also anomaly-free since they satisfy the above constraints (at least at face level), but there is no string theory with these gauge groups!
Why is this the case? Have there been any modern developments in a rigorous proof of their absence?
